MATH REFRESHER
FOR SCIENTISTS
AND ENGINEERS
Third Edition
JOHN R. FANCHI
A JOHN WILEY & SONS, INC., PUBLICATION
MATH REFRESHER
FOR SCIENTISTS
AND ENGINEERS
MATH REFRESHER
FOR SCIENTISTS
AND ENGINEERS
Third Edition
JOHN R. FANCHI
A JOHN WILEY & SONS, INC., PUBLICATION
Copyright # 2006 by John Wiley & Sons, Inc. All rights reserved
Published by John Wiley & Sons, Inc., Hoboken, New Jersey
Published simultaneously in Canada
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Library of Congress Cataloging-in-Publication Data.
Fanchi, John R.
Math refresher for scientists and engineers/John R. Fanchi.—3rd ed.
p. cm.
Includes bibliographical references and index.
ISBN-13: 978-0-471-75715-3
ISBN-10: 0-471-75715-2
1. Mathematics. I. Title
QA37.2.F35 2006
5120 .1—dc22
2005056262
Printed in the United States of America
10 9 8
7 6 5
4 3 2 1
CONTENTS
PREFACE
xi
1
1
ALGEBRA
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
2
GEOMETRY, TRIGONOMETRY,
AND HYPERBOLIC FUNCTIONS
2.1
2.2
2.3
2.4
2.5
3
Algebraic Axioms / 1
Algebraic Operations / 6
Exponents and Roots / 7
Quadratic Equations / 9
Logarithms / 10
Factorials / 12
Complex Numbers / 13
Polynomials and Partial Fractions / 17
Geometry / 21
Trigonometry / 26
Common Coordinate Systems / 32
Euler’s Equation and Hyperbolic Functions / 34
Series Representations / 37
ANALYTIC GEOMETRY
3.1
21
41
Line / 41
vii
viii
CONTENTS
3.2
3.3
4
LINEAR ALGEBRA I
4.1
4.2
4.3
5
7
8
9
10
117
Line Integral / 117
Double Integral / 119
Fourier Analysis / 121
Fourier Integral and Fourier Transform / 124
Time Series and Z Transform / 127
Laplace Transform / 130
ORDINARY DIFFERENTIAL EQUATIONS
10.1
107
Indefinite Integrals / 107
Definite Integrals / 109
Solving Integrals / 112
Numerical Integration / 114
SPECIAL INTEGRALS
9.1
9.2
9.3
9.4
9.5
9.6
93
Partial Differentiation / 94
Vector Analysis / 96
Analyticity and the Cauchy – Riemann Equations / 103
INTEGRAL CALCULUS
8.1
8.2
8.3
8.4
79
Limits / 79
Derivatives / 82
Finite Difference Concept / 87
PARTIAL DERIVATIVES
7.1
7.2
7.3
65
Vectors / 65
Vector Spaces / 69
Eigenvalues and Eigenvectors / 71
Matrix Diagonalization / 74
DIFFERENTIAL CALCULUS
6.1
6.2
6.3
51
Rotation of Axes / 51
Matrices / 53
Determinants / 61
LINEAR ALGEBRA II
5.1
5.2
5.3
5.4
6
Conic Sections / 44
Polar Form of Complex Numbers / 48
First-Order ODE / 133
133
CONTENTS
10.2
10.3
10.4
11
12
13.3
13.4
13.5
14
15
16
203
Contravariant and Covariant Vectors / 203
Tensors / 207
The Metric Tensor / 210
Tensor Properties / 213
PROBABILITY
16.1
16.2
191
Calculus of Variations with One Dependent Variable / 191
The Beltrami Identity and the Brachistochrone Problem / 195
Calculus of Variations with Several Dependent Variables / 198
Calculus of Variations with Constraints / 200
TENSOR ANALYSIS
15.1
15.2
15.3
15.4
181
Classification / 181
Integral Equation Representation
of a Second-Order ODE / 182
Solving Integral Equations: Neumann Series Method / 185
Solving Integral Equations with Separable Kernels / 187
Solving Integral Equations with Laplace Transforms / 188
CALCULUS OF VARIATIONS
14.1
14.2
14.3
14.4
167
Boundary Conditions / 168
PDE Classification Scheme / 168
Analytical Solution Techniques / 169
Numerical Solution Methods / 175
INTEGRAL EQUATIONS
13.1
13.2
151
Higher-Order ODE with Constant Coefficients / 151
Variation of Parameters / 155
Cauchy Equation / 157
Series Methods / 158
Laplace Transform Method / 164
PARTIAL DIFFERENTIAL EQUATIONS
12.1
12.2
12.3
12.4
13
Higher-Order ODE / 141
Stability Analysis / 142
Introduction to Nonlinear Dynamics and Chaos / 145
ODE SOLUTION TECHNIQUES
11.1
11.2
11.3
11.4
11.5
ix
Set Theory / 219
Probability Defined / 222
219
x
CONTENTS
16.3
16.4
17
PROBABILITY DISTRIBUTIONS
17.1
17.2
17.3
17.4
18
245
Probability and Frequency / 245
Ungrouped Data / 247
Grouped Data / 249
Statistical Coefficients / 250
Curve Fitting, Regression, and Correlation / 252
SOLUTIONS TO EXERCISES
S1
S2
S3
S4
S5
S6
S7
S8
S9
S10
S11
S12
S13
S14
S15
S16
S17
S18
229
Joint Probability Distribution / 229
Expectation Values and Moments / 233
Multivariate Distributions / 235
Example Probability Distributions / 240
STATISTICS
18.1
18.2
18.3
18.4
18.5
19
Properties of Probability / 222
Probability Distribution Defined / 227
257
Algebra / 257
Geometry, Trigonometry, and Hyperbolic Functions / 262
Analytic Geometry / 267
Linear Algebra I / 269
Linear Algebra II / 276
Differential Calculus / 280
Partial Derivatives / 286
Integral Calculus / 290
Special Integrals / 293
Ordinary Differential Equations / 299
ODE Solution Techniques / 306
Partial Differential Equations / 312
Integral Equations / 316
Calculus of Variations / 323
Tensor Analysis / 327
Probability / 331
Probability Distributions / 333
Statistics / 337
REFERENCES
339
INDEX
343
PREFACE
Math Refresher for Scientists and Engineers, Third Edition is intended for
people with technical backgrounds who would like to refresh their math
skills. This book is unique because it contains in one source an overview of the
essential elements of a wide range of mathematical topics that are normally found
in separate texts. The first edition began with relatively simple concepts in
college algebra and trigonometry and then proceeded to more advanced concepts
ranging from calculus to linear algebra (including matrices) and differential
equations. Numerical methods were interspersed throughout the presentation.
The second edition added chapters that discussed probability and statistics. In this
third edition, three new chapters with exercises and solutions have been added.
The new material includes chapters on integral equations, the calculus of variations,
and tensor analysis. Furthermore, the discussion of integral transforms has been
expanded, a section on partial fractions has been added, and several new exercises
have been included.
Math Refresher for Scientists and Engineers, Third Edition is designed for the
adult learner and is suitable for reference, self-review, adult education, and
college review. It is especially useful for the professional who wants to understand
the latest technology, the engineer who is preparing to take a professional engineering exam, or students who wish to refresh their math background. The focus of the
book is on modern, practical applications and exercises rather than theory.
Chapters are organized to include a review of important principles and methods.
Interwoven with the review are examples, exercises, and applications. Examples
are intended to clarify concepts. Exercises are designed to make you an active
participant in the review process. Keeping in mind that your time is valuable, I
have restricted the number of exercises to a number that should complement and
xi
XII
PREFACE
supplement the text while minimizing nonessential repetition. Solutions to exercises
are separated from the exercises to allow for a self-paced review.
Applications provide an integration of concepts and methods from one or more
sections. In many cases, they introduce you to modern topics. Applications may
include material that has not yet been covered in the book, but should be familiar
if you are using the book as a “refresher course.” You may wish to return to the
applications after a first reading of the rest of the text.
I developed much of the material in this book as course notes for continuing education courses in Denver and Houston. I would like to thank Kathy Fanchi, Cindee
Calton, Chris Fanchi, and Stephanie Potter for their assistance in the development of
this material. Any written comments or suggestions for improving the material are
welcome.
JOHN R. FANCHI
CHAPTER 1
ALGEBRA
Many practical applications of advanced mathematics assume the practitioner is
fluent in the language of algebra. This review of algebra includes a succinct discussion of sets and groups, as well as a presentation of basic operations with both real
and complex numbers. Although much of this material may appear to be elementary
to the reader, its presentation here establishes a common terminology and notation
for later sections.
1.1
ALGEBRAIC AXIOMS
Sets
A set is a collection of objects. The objects are called elements or members of the set.
Example: The expression “a and b belong to the set A” can be written a, b [ A,
where a, b are elements or members of set A.
Let iff be the abbreviation for “if and only if.” Two sets A, B are equal iff they have
the same members. In other words, sets A, B satisfy the equality A ¼ B iff set A has
the same members as set B.
Math Refresher for Scientists and Engineers, Third Edition By John R. Fanchi
Copyright # 2006 John Wiley & Sons, Inc.
1
2
MATH REFRESHER FOR SCIENTISTS AND ENGINEERS
Example: Let A ¼ {2, 3, 4} and B ¼ {4, 5, 6}; then 3 [ A but 3 Ó B, where Ó
denotes “does not belong to” or “is not a member of.” We conclude that set A
and set B are not equal.
The union of two sets A and B (A < B) is the set of all elements that belong to A or
to B or to both. The intersection of two sets A and B (A > B) is the set of all elements
that belong to both A and B.
Example: Let A ¼ {2, 3, 4} and B ¼ {4, 5, 6}; then A < B ¼ {2, 3, 4, 5, 6} and
A > B ¼ {4}.
Union and intersection may be depicted by a Venn diagram (Figure 1.1). All the
elements of a set are enclosed in a circle. The union of the sets is the total area
bounded by the circles in the Venn diagram. The intersection is the overlapping
cross-hatched area.
The null set or empty set is the set with no elements and is denoted by 1.
The set of Real numbers R includes rational numbers and irrational numbers. A
rational number is a number that may be written as the quotient of two integers. All
other real numbers are irrational numbers. A real number classification scheme is
shown as follows:
Rational numbers {a=bja, b [ integers, b = 0}
Noninteger rationals (e.g., 12, 13, . . .)
Integer rationals f. . . , 22, 21, 0, 1, 2, . . .g
Negative integers f. . . , 23, 22, 21g
Zero f0g
Natural numbers f1, 2, 3, . . .g
Irrational numbers
A function f from a set A to a set B assigns to each element x [ A a unique
element y [ B. This may be written f : A ! B and the function f is said to be a
Figure 1.1
Venn diagram.
ALGEBRA
3
map of elements from set A to set B. The element y is the value of f at x and is written
y ¼ f (x). The set A is the domain of f, and x is an element in the domain of f. The set
B is the range of f, and the value of y is in the range of f.
Example: The equation y ¼ x2 is a function that maps the element x [ R to the
element y [ R: The element y is a function of x, which is often written as
y ¼ f (x). By contrast, the equation y ¼ x2 does not uniquely define x as a function
of y because x is not single valued; that is, y ¼ 4 is associated with both x ¼ 2
and x ¼ À2. For further discussion of functions, see texts such as Swokowski
[1986] or Riddle [1992].
Operations
pffiffiffi
Square Root For a [ R, where a is positive or zero, the notation a signifies the
square
proot
ffiffiffipffiffiffiof a. It is the nonnegative
pffiffiffi real number whose product with itself gives a;
thus a a ¼ a. The notation a is sometimes called p
theffiffiffi principal square root
[Stewart et al., 1992]. The negative square root of a is À a.
pffiffiffi
Example: Let a ¼ 169; then paffiffiffiffiffiffiffi
is ffi13 since 13 Â 13 ¼ 169: The number a ¼ 169
has the negative square root À 169 ¼ À13.
Absolute Value For all a [ R, if a is positive or zero, then jaj ¼ a; if a is negative, then jaj ¼ Àa.
Example: Let a ¼ {5, À1, À6, À14:3}; then jaj ¼ {5, 1, 6, 14:3}.
Greater Than Elements can be ordered using the concepts of greater than
and less than. For all a, b [ R, b is less than a if for some positive
real number c we have a ¼ b ỵ c. For such a condition, a is said to be greater
than b, which may be written as a . b. Alternatively, a is greater than b if
a À b . 0. Similarly, we may say that b is less than a and write it as b , a if
a À b . 0.
Example: Let a ¼ 3 and b ¼ 22. Is a greater than b? To satisfy b ỵ c ẳ a, we must
have c ẳ 5. Since c is a positive real number, we know that a . b.
Groups
Many of the axioms presented below will seem obvious, particularly when thought
of in terms of the set of real numbers. The validity of the axioms becomes much less
obvious when applied to different sets, such as sets of matrices.
4
MATH REFRESHER FOR SCIENTISTS AND ENGINEERS
Axioms A binary operation ab between a pair of elements a, b [ G exists if
ab [ G. Let G be a nonempty set with a binary operation. G is a group if the
following axioms hold:
[G1]
[G2]
[G3]
Associative law: For any a, b, c [ G, (ab)c ¼ a(bc).
Identity element: There exists an element e [ G such that ae ¼ ea ¼ a
for every a [ G.
Inverse: For each a [ G there exists an element a 21 [ G such that
aa 21 ¼ a 21 a ¼ e.
Example: Let a ¼ 3, b ¼ 2, and c ¼ 5. We verify the associative law for multiplication by example:
(ab)c ¼ a(bc)
(3 Á 2) Á 5 ¼ 3 Á (2 Á 5)
6 Á 5 ¼ 3 Á 10
30 ¼ 30
The identity element for multiplication of real numbers is 1, and the inverse of a
real number a is 1/a as long as a = 0. The inverse of a ¼ 0 is undefined.
Abelian Groups A group G is an abelian group if ab ¼ ba for every a, b [ G.
The elements of the group are said to commute. The commutativity of the product of
two elements a, b may be written
ẵa, b ; ab ba ẳ 0
Commutativity is always satisfied for the set of real numbers but is often not satisfied
for matrices.
Example: Let a and b be the matrices
a¼
1
,
0
0
1
b¼
1
0
0 À1
Then
ab ¼
0
1
1
0
1
0
0
À1
¼
0 À1
1
0
ALGEBRA
ba ¼
1
0
0
À1
0
1
0
1
¼
À1
0
1
0
5
Therefore matrices a and b do not commute; that is, ab = ba.
Hint: To perform the multiplication of the above matrices, recall that
a11
a21
a12
a22
b11
b21
b12
b22
a b ỵ a12 b21
ẳ 11 11
a21 b11 ỵ a22 b21
a11 b12 ỵ a12 b22
a21 b12 þ a22 b22
where faijg and fbijg are elements of the matrices a and b, respectively. An analogous
relation applies when the order of multiplication changes from ab to ba. For more
discussion of matrices, see Chapter 4.
Rings
Axioms
Let R be a nonempty set with two binary operations:
1. Addition (denoted by ỵ), and
2. Multiplication (denoted by juxtaposition).
The nonempty set R is a ring if the following axioms hold:
[R1]
[R2]
[R3]
[R4]
[R5]
[R6]
Associative law of addition: For any a, b, c [ R, (a ỵ b) ỵ c ẳ a ỵ (b ỵ c):
Zero element: There exists an element 0 [ R such that a ỵ 0 ẳ 0 þ a ¼
a for every a [ R.
Negative of a: For each a [ R there exists an element 2a [ R such that
a ỵ (a) ẳ (a) ỵ a ¼ 0.
Commutative law of addition: For any a, b [ R, a ỵ b ẳ b ỵ a.
Associative law of multiplication: For any a, b, c [ R, (ab)c ¼ a(bc).
Distributive law: For any a, b, c [ R, we have
(i) a(b ỵ c) ẳ ab ỵ ac, and
(ii) (b þ c)a ¼ ba þ ca.
Subtraction is defined in R by a b ; a ỵ (b). R is a commutative ring if
ab ¼ ba for every a, b [ R. R is a ring with a unit element if there exists a
nonzero element 1 [ R such that a . 1 ¼ 1 . a ¼ a for every a [ R.
Integral Domain and Field A commutative ring R with a unit element is an integral domain if R has no zero divisors, that is, if ab ¼ 0 implies a ¼ 0 or b ¼ 0. A commutative ring R with a unit element is a field if every nonzero a [ R has a
multiplicative inverse; that is, there exists an element a 21 [ R such that
aa 21 ¼ a 21a ¼ 1.
6
MATH REFRESHER FOR SCIENTISTS AND ENGINEERS
1.2
ALGEBRAIC OPERATIONS
Algebraic operations may be presented as a collection of axioms. In all cases assume
a, b, c, d [ R. The following presents equality axioms:
EQUALITY AXIOMS
Reflexive law
Symmetric law
Transitive law
Substitution law
a¼a
If a ¼ b, then b ¼ a
If a ¼ b and b ¼ c, then a ¼ c
If a ¼ b, then a may be substituted
for b or b for a in any expression
Ordering relations obey the following axioms:
ORDER AXIOMS
Trichotomy law
Exactly one of the following is
true: a , b, a ¼ b, or a . b
If a , b and b , c, then a , c
If a,b . 0, then a ỵ b . 0 and
ab . 0
Transitive law
Closure for positive
numbers
Axioms for addition and multiplication operations are summarized as follows:
ADDITION AXIOMS
Closure law for addition
Commutative law for addition
Associative law for addition
Identity law of addition
Additive inverse law
aỵb [R
aỵb ẳ bỵa
(a ỵ b) ỵ c ẳ a þ (b þ c)
aþ0 ¼ 0þa¼ a
a þ (2a) ¼ (2a) ỵ a ẳ 0
MULTIPLICATION AXIOMS
Closure law for multiplication
Commutative law for multiplication
Associative law for multiplication
Identity law of multiplication
Multiplication inverse law
Distributive law
ab [ R
ab ¼ ba
(ab)c ¼ a(bc)
a.1 ¼ 1.a ¼ a
a . (1/a) ¼ (1/a) . a ¼ 1 for a = 0
a . (b ỵ c) ẳ a . b ỵ a . c
ALGEBRA
7
Several algebraic properties follow from the axioms. Some of the most useful are
as follows:
MISCELLANEOUS ALGEBRAIC PROPERTIES
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
a . 0 ¼ 0.
2(2a) ¼ a.
2a ¼ 21 . a.
If a ¼ b, then a ỵ c ẳ b ỵ c.
If a ỵ c ẳ b ỵ c, then a ẳ b.
If a ẳ b, then a . c ¼ b . c.
If a .c ¼ b . c, then a ¼ b for c = 0.
a 2 b ẳ a ỵ (2b).
a/b ẳ c/d if and only if a . d ¼ b . c for b, d = 0.
a/b ¼ (a . c)/(b . c) for b, c = 0.
(a/c) ỵ (b/c) ẳ (a ỵ b)/c for c = 0.
(a/b) . (c/d) ¼ (a . c)/(b . d) for b, d = 0.
For a more detailed discussion of algebraic axioms and properties, see such references as Swokowski [1986], Breckenbach et al. [1985], Gustafson and Frisk [1980],
and Rich [1973].
1.3
EXPONENTS AND ROOTS
Exponents
Exponents obey the three laws tabulated as follows:
EXPONENTS
Products
Quotient
Power
a m . a n ¼ a mỵn
am
ẳ amn if m . n
an
am
ẳ 1 if m ¼ n
an
m
a
1
¼
if m , n
an anÀm
(a m)n ¼ a mn
The number a raised to a negative power is given by a 2m ¼ 1/a m. Any nonzero real
number raised to the power 0 equals 1; thus a 0 ¼ 1 if a = 0. In the case of the
number 0, we have the exponential relationships 00 ¼ 0, 0x ¼ 0 for all x.
8
MATH REFRESHER FOR SCIENTISTS AND ENGINEERS
Example: Scientific notation illustrates the use of exponentiation. In particular,
numbers such as 86,400 and 0.00001 can be written in the form 8.64 Â 104 and
1 Â 1025, respectively. Properties of exponents are then used to perform
calculations; thus
(86,400)(0:00001) ¼ (8:64 Â 104 )(1 Â 10À5 )
¼ (8:64 Â 1)(104 Â 10À5 )
¼ (8:64)(104À5 )
¼ 8:64 Â 10À1 ¼ 0:864
Scientific notation is a means of compactly writing very large or very small
numbers. It is also useful for making order or magnitude estimates. In this case,
numbers are written so that they can be rounded off to approximately 1 Â 10n,
and then exponents are combined to estimate products or quotients. For example,
the number 86,400 is approximated as 8.64 Â 104 ¼ 0.864 Â 105 ¼ 1 Â 105 so that
(86,400)(0:00001) % (1 Â 105 )(1 Â 10À5 ) ¼ 1
as expected.
Roots
The solution of the equation
bn ¼ a
may be written formally as the n th root of a equals b, or
ffiffiffi
p
n
a¼b
A real number raised to a fraction p/q may be written as the q th root of a p, or
pffiffiffiffiffi
a p=q ¼ q ap
A summary of relations for roots is as follows:
ROOTS
Product
Quotient
Power
pffiffiffi
a1=x ¼ x a
p
ffiffiffiffiffi
pffiffiffipffiffiffi
x
ab ¼ x a x b ¼ a1=x b1=x
rffiffiffi p
ffiffiffi
x
a a1=x
x a
ffiffiffi ¼ 1=x
¼p
x
b
b b
pffiffiffiffiffi
(a1=y )x ¼ ax=y ¼ y ax
p
ffiffiffiffiffiffiffiffiffi
p
x y
a ¼ (a1=y )1=x ¼ a1=xy
ALGEBRA
1.4
9
QUADRATIC EQUATIONS
The solutions of the quadratic equation
ax2 ỵ bx ỵ c ¼ 0
are
x+ ¼
Àb +
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
b2 À 4ac
2a
where x+ is the solution when the square root term is added and x2 is the solution
when the square root term is subtracted. The quadratic equation is an example of
a polynomial equation. Any polynomial equation with a term x n, where the
degree n is the largest positive integer, has n solutions. The quadratic equation
has degree n ¼ 2 and has two solutions. Cubic equations, which are often encountered in chemical equations of state, have degree n ¼ 3 and have three solutions.
The solutions are not necessarily real. Polynomial equations are discussed in
more detail in Section 1.8.
Example: The solutions of 2x 2 2 6x ỵ 4 ẳ 0 are
p
( 6)2 4(2) 4
ẳ2
xỵ ẳ
2(2)
p
6 ( 6)2 4(2) 4
x ẳ
ẳ1
2(2)
6ỵ
These solutions can be used to factor the quadratic equation into a product of terms
that have degree n = 1; thus
2x2 À 6x þ 4 ¼ 2(x À 1)(x À 2) ¼ 0
EXERCISE 1.1: Given ax 2 ỵ bx ỵ c ẳ 0, find x in terms of a, b, c.
The following exercise may be solved using elementary algebraic operations
from the preceding sections.
EXERCISE 1.2: (a) Expand and simplify (x ỵ y)2 , (x y)2 , (x ỵ y)3 , (x ỵ y þ z)2 ,
(ax þ b) Á (cx þ d), and (x ỵ y)(x y). (b) Factor 3x3 ỵ 6x2 y ỵ 3xy2 .
10
1.5
MATH REFRESHER FOR SCIENTISTS AND ENGINEERS
LOGARITHMS
Definition
The logarithm of the real number x . 0 to the base a is written as loga x and is
defined by the relationship:
If x ¼ ay then y ¼ loga x
Logarithms are the inverse operation of exponentiation and obey the following
three laws:
LOGARITHMS
Product
Quotient
Power
loga(xy)
¼ loga x þ loga y
x
loga
¼ loga x À loga y
y
n
loga (x ) ¼ nloga x
loga (1) ¼ 0
logx (x) ¼ 1
Logarithms to the base e % 2:71828 are called natural logarithms and are written
as ln x ; loge x. The equation for changing the base of the logarithm from base a to
base b is logb x ¼ loga x= loga b.
EXERCISE 1.3: Let y ¼ loga x: Change the base of the logarithm from base a to
base b, then set a ¼ 10 and b ¼ 2:71828 % e and simplify.
Application: Fractals. Benoit Mandelbrot [1967] introduced the concept of
fractional dimension, or fractal, to describe the complexities of geographical
curves. One of the motivating factors behind this work was an attempt to determine
the lengths of coastlines. This problem is discussed here as an introduction to
fractals.
We can express the length of a coastline Lc by writing
Lc ¼ N 1
(1:5:1)
where N is the number of measurement intervals with fixed length 1: For example, 1
could be the length of a meter stick. Geographers have long been aware that the
length of a coastline depends on its regularity and the unit of length 1: They
found that Lc for an irregular coastline, such as the coast of Australia
(Figure 1.2), increases as the scale of measurement 1 gets shorter. This behavior
is caused by the ability of the smaller measurement interval to more accurately
include the lengths of irregularities in the measurement of Lc :
